1. Field of the Invention
The present invention relates to a data conversion method based on scale-adjusted β-map, and particularly to an A/D converter and a chaos generator using a discrete time integrator for adapting to an integrated circuit.
2. Description of the Related Art
Conventionally, the data conversion method based on β-map has been proposed (see Non-Patent Document 1 listed below). This method is, as compared to the PCM (Pulse Code Modulation) method, superior in terms of stability as analog circuits. In other words, in the PCM method, the circuit operation potentially diverges due to the fluctuation or noises of circuit parameters, such as a threshold of a quantizer, a gain constant of a multiply-by-two amplifier. However, the convergence of conversion errors for bit lengths is exponential. On the other hand, in the ΣΔ-type converters, while the circuit operation is stable, the oversampling or the like is required to improve the conversion accuracy. In addition, they take more bit lengths for the convergence of conversion errors. In contrast, the data conversion method based on β-map is robust for mismatches of circuit parameters similar to the ΣΔ-type converters, while it provides the practically optimum rate-distortion characteristics similar to the PCM method.
In recent years, decoding algorithms have been disclosed for minimizing the errors using interval analysis, along with the design principles of circuit parameters, in order to further improve the performance of the data conversion method based on β-map (see Patent Documents 1 and 2, and Non-Patent Documents 2 to 4 listed below). Moreover, in order to enhance the degree of freedom of the circuit design, the data conversion method based on scale-adjusted β-map has been proposed that allows the gain β of an amplifier circuit and the threshold of a quantizer to be set up independently (see Patent Document 2, and Non-Patent Documents 3 and 4 listed below). In this regard, this method includes a conventional data converter based on β-map in certain cases.
In Patent Documents 1 and 2, and Non-Patent Documents 1 to 4 listed below, the block diagrams are presented to configure data encoders based on β-map (hereinbelow, A/D converters), the A/D converters based on scale-adjusted β-map, or the like. However, these block diagrams are not suitable for actual implementation of circuits, particularly integrated circuits.
Now, detailed description will be provided.
The scale-adjusted β-map S (•) is described in Equation (1) (see Patent Document 2, and Non-Patent Documents 3 and 4).
                              S          ⁡                      (            x            )                          =                  {                                                                                          β                    ⁢                                                                                  ⁢                    x                                    ,                                                                              x                  ∈                                      [                                          0                      ,                                              γ                        ⁢                                                                                                  ⁢                        v                                                              )                                                                                                                                                                  β                      ⁢                                                                                          ⁢                      x                                        -                                          s                      ⁡                                              (                                                  β                          -                          1                                                )                                                                              ,                                                                              x                  ∈                                      [                                                                  γ                        ⁢                                                                                                  ⁢                        v                                            ,                      s                                        )                                                                                                          (        1        )            wherein, υε[s(β−1), s) is a threshold parameter, 1<β<2 is a conversion radix, and γ=1/β, s>0 is a scaling constant. In addition, when s=(β−1)−1, the scale-adjusted β-map S (•) is the same as the β-map C(•) below (see Non-Patent Document 1).
                              C          ⁡                      (            x            )                          =                  {                                                                                          β                    ⁢                                                                                  ⁢                    x                                    ,                                                                              x                  <                                      γ                    ⁢                                                                                  ⁢                    v                                                                                                                                                                  β                      ⁢                                                                                          ⁢                      x                                        -                    1                                    ,                                                                              x                  ≥                                      γ                    ⁢                                                                                  ⁢                    v                                                                                                          (        2        )            wherein, υε[1, (β−1)−1). Also, when s=β•(β−1)−1, the scale-adjusted β-map S (•) of Equation (1) above is the same as another β-map D(•) (see Non-Patent Document 1).
                              D          ⁡                      (            x            )                          =                  {                                                                                          β                    ⁢                                                                                  ⁢                    x                                    ,                                                                              x                  <                                      γ                    ⁢                                                                                  ⁢                    v                                                                                                                                            β                    ⁡                                          (                                              x                        -                        1                                            )                                                        ,                                                                              x                  ≥                                      γ                    ⁢                                                                                  ⁢                    v                                                                                                          (        3        )            wherein, υε[β, β(β−1)−1).
Assuming that the discrete time is tn (n is a natural number), and then by using it to rewrite Equation (1) above as the one-dimensional discrete time dynamical system, it can be described as
                              x          ⁡                      (                          t                              n                +                1                                      )                          =                              S            ⁡                          (                              x                ⁡                                  (                                      t                    n                                    )                                            )                                =                      {                                                                                                      β                      ⁢                                                                                          ⁢                                              x                        ⁡                                                  (                                                      t                            n                                                    )                                                                                      ,                                                                                                              x                      ⁡                                              (                                                  t                          n                                                )                                                              ∈                                          [                                              0                        ,                                                  γ                          ⁢                                                                                                          ⁢                          v                                                                    )                                                                                                                                                                                      β                        ⁢                                                                                                  ⁢                                                  x                          ⁡                                                      (                                                          t                              n                                                        )                                                                                              -                                              s                        ⁡                                                  (                                                      β                            -                            1                                                    )                                                                                      ,                                                                                                              x                      ⁡                                              (                                                  t                          n                                                )                                                              ∈                                          [                                                                        γ                          ⁢                                                                                                          ⁢                          v                                                ,                        s                                            )                                                                                                                              (        4        )            An example of the one-dimensional map is shown in FIG. 12. In this figure, β=5/3, s=3, υ=5/2, and γυ=3/2. FIG. 12 also illustrates the trajectory with x(t1)=0.6 as an initial value. As shown in FIG. 12, the trajectory is resultantly confined within an invariant subinterval [υ−s(β−1), υ) (the region D in FIG. 12).
Moreover, a binary variable b(tn)ε{0,1} is defined as
                              b          ⁡                      (                          t              n                        )                          =                                            Q              θ                        ⁡                          (                              x                ⁡                                  (                                      t                    n                                    )                                            )                                =                      {                                                                                0                    ,                                                                                                              x                      ⁡                                              (                                                  t                          n                                                )                                                              ∈                                          [                                              0                        ,                        θ                                            )                                                                                                                                        1                    ,                                                                                                              x                      ⁡                                              (                                                  t                          n                                                )                                                              ∈                                          [                                              θ                        ,                        s                                            )                                                                                                                              (        5        )            wherein, Qθ(•) is a quantizer with θ as the threshold. Also, the following applies hereinbelow.θ=γυ  (6)At this time, Equation (4) above can be described asx(tn+1)=βx(tn)−b(tn)s(β−1)  (7)
Here, assume that the input signal xinput is sampled at t=t1. That is,x(t1)=xinput  (8)At this time, by repeating Equation (7) above from t=t1 to t=tL (L is a bit length after A/D conversion), the binary signal train BS(xinput) corresponding to the input signal xinput is obtained.BS(xinput)=(b1b2 . . . bL)β,s  (9)wherein, bn=b(tn) (n=1, 2, . . . , L), bL=b(tL) is the LSB (least significant bit), and b1=b(t1) is the MSB (most significant bit).
Here, the tolerance σv, of the threshold parameter υ of the quantizer Qθ(•) is given as follows with s and β (see Patent Document 2, and Non-Patent Documents 3 and 4).σv=s(2−β)  (10)This is shown in FIG. 12 as the bold line on the axis x(tn+1) Accordingly, the threshold θ of the quantizer is allowed to vary within the range described as follows (see Patent Document 2, and Non-Patent Documents 3 and 4).σθ=γσv=γs(2−β)=s(2γ−1)  (11)This is shown in FIG. 12 as the bold line on the axis x(tn).
The configuration diagrams of the A/D converter using the scale-adjusted β-map are shown in Patent Document 2, and Non-Patent Documents 3 and 4 listed below. FIG. 13 shows a configuration diagram of the A/D converter using the scale-adjusted β-map. However, the as-is configuration is not suitable for integrated circuits.                Patent Document 1: WO 2009/014057        Patent Document 2: WO 2010/024196        Non-Patent Document 1: I. Daubechies, R. A. DeVore, C. S. Gunturk, and V. A. Vaishampayan, “A/D conversion with imperfect quantizers”, IEEE Transactions on Information Theory, Vol. 52, No. 3, pp. 874-885, 2006        Non-Patent Document 2: S. Hironaka, T. Kohda, and K. Aihara, “Markov chain of binary sequences generated by A/D conversion using β-encoder”, in Proceedings of IEEE Workshop on Nonlinear Dynamics of Electronic Systems, pp. 261-264, Tokushima, Japan, 2007        Non-Patent Document 3: S. Hironaka, T. Kohda, and K. Aihara, “Negative β-encoder”, in Proceedings of International Symposium on Nonlinear Theory and Its Applications, pp. 564-567, Budapest, Hungary, 2008        Non-Patent Document 4: T. Kohda, S. Hironaka, and K. Aihara, “Negative β-encoder”, Preprint, arXiv:0808.2548v2[cs.IT], 28 Jul. 2009, http://arxiv.org/abs/0808.2548        